An empirical analysis of the relationship between the hedge ratio and hedging horizon: A simultaneous estimation of the short- and long-run hedge ratios

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OF THE

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ELATIONSHIP

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ETWEEN THE

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EDGE

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ATIO AND

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EDGING

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ORIZON

: A S

IMULTANEOUS

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STIMATION OF THE

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HORT

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AND

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ONG

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UN

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EDGE

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ATIOS

SHENG-SYAN CHEN* CHENG-FEW LEE KESHAB SHRESTHA

This article analyzes the effects of the length of hedging horizon on the optimal hedge ratio and hedging effectiveness using 9 different hedging horizons and 25 different commodities. We discuss the concept of short-and long-run hedge ratios short-and propose a technique to simultaneously

We are grateful to an anonymous referee and the Editor for their helpful comments and suggestions. The usual caveat of course applies.

*Correspondence author, Department of Finance, College of Management, Yuan Ze University, 135 Yuan-Tung Road, Chung-Li, Taoyuan, Taiwan; e-mail: fnschen@saturn.yzu.edu.tw

Received December 2002; Accepted April 2003

■ Sheng-Syan Chen is a professor in the Department of Finance, College of Management, Yuan Ze University, in Taoyuan, Taiwan.

■ Cheng-Few Lee is a distinguished professor in the Department of Finance and Economics, School of Business, Rutgers University in Piscataway, New Jersey and Graduate Institute of Finance at National Chiao Tung University in Taiwan.

■ Keshab Shrestha is an associate professor, Division of Banking and Finance, Nanyang Business School, Nanyang Technological University, Singapore.

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360 Chen, Lee, and Shrestha

estimate them. The empirical results indicate that the short-run hedge ratios are signiﬁcantly less than 1 and increase with the length of hedging horizon. We also ﬁnd that hedging effectiveness increases with the length of hedging horizon. However, the long-run hedge ratio is found to be close to the naïve hedge ratio of unity. This implies that, if the hedging horizon is long, then the naïve hedge ratio is close to the optimum hedge ratio. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:359–386, 2004

INTRODUCTION

One of the best uses of derivative securities, such as futures contracts, is in hedging. In the past, both academicians and practitioners have shown great interest in the issue of hedging with futures, which is evident from a large number of articles written in this area. Most of the studies on the hedge ratio deal with either (a) the derivation of the optimal hedge ratio based on different objective functions, or (b) the empirical estimation of the optimal hedge ratio. Some of the derivations of the optimal hedge ratio are based on the minimization of return variance or maximization of the expected utility. Other derivations of the optimal hedge ratio are based on the mean-Gini coefﬁcient and generalized semivariance. A brief discussion on this can be found in Chen, Lee, and Shrestha (2001).

Studies that deal with the empirical estimation of the optimal hedge ratio employ many different techniques, ranging from simple to complex ones. For example, some of the studies use such a simple method as the ordinary least-squares (OLS) technique (e.g., see Benet, 1992; Ederington, 1979; Malliaris & Urrutia, 1991). However, others use more complex methods such as the conditional heteroscedastic (ARCH or GARCH) method (e.g., see Baillie & Myers, 1991; Cecchetti, Cumby, & Figlewski, 1988; Sephton, 1993), the random coefficient method (e.g., see Grammatikos & Saunders, 1983), the co-integration method (e.g., see Chou, Fan, & Lee, 1996; Geppert, 1995; Ghosh, 1993; Lien & Luo, 1993), and the co-integration–heteroscedastic method (e.g., see Kroner & Sultan, 1993).

Most of the empirical studies, however, ignore the effect of hedging horizon length on the optimal hedge ratio and hedging effectiveness.1_A few studies that consider the effect of the length of hedging horizon include Ederington (1979), Hill and Schneeweis (1982), Malliaris and Urrutia (1991), Benet (1992), and Geppert (1995). These studies ﬁnd

1_{Lee and Leuthold (1983) found that return characteristics depend on the investment horizon.}

Therefore, we expect that the investment horizon will have an impact on the optimal hedge ratio as well.

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that in-sample hedging effectiveness tends to increase as the investment horizon lengthens. However, all of these studies, except Geppert (1995), consider 2–3 different hedging horizon lengths, whereas only Geppert (1995) considers 12 different hedging horizon lengths.2

From the estimates of the optimal hedge ratios reported in these studies, we can see that the optimal hedge ratio tends to increase with the length of hedging horizon. It is interesting to note that, even though all of these studies report the optimal hedge ratios for different hedging hori-zons, only Geppert (1995) analyzes the relationship between the optimal hedge ratio and the length of hedging horizon. It is important to note that all these studies consider the minimum variance (MV) hedge ratio, instead of the other hedge ratios based on expected utility, extended mean-Gini coefﬁcient, and generalized semivariance.

This article examines the effects of hedging horizon length on the optimal hedge ratio and hedging effectiveness in greater detail, using 25 different futures contracts and 9 different hedging horizons.3 _{Based on} the relationship between the hedge ratio and the length of hedging hori-zon, we discuss the concepts of short- and long-run hedge ratios and pro-pose a method that can be used to simultaneously estimate the short-and long-run hedge ratios. We consider only the MV hedge ratio based on the considerations as discussed below. First, as mentioned above, most of the existing studies analyze the MV hedge ratio. In order to com-pare our results with those in existing studies, we also consider the MV hedge ratio. Second, the MV hedge ratio is the most heavily used, ana-lyzed, and discussed hedge ratio. Finally, it can be shown that, under some normality and martingale conditions, most of the hedge ratios based on other criteria (e.g., expected utility, extended mean-Gini coefﬁ-cient, and generalized semivariance) converge to the MV hedge ratio.4

It is important to note that our analysis differs from the one done by Geppert (1995) (hereafter referred to as JG), in that the model used by JG requires both the spot and futures prices to have a single unit root. Therefore, the hedge ratio derived by JG is valid only if the unit-root condition is satisfied. However, we find that 14 out of 25 different

2_{Actually, the analytical expression for the optimal hedge ratio [equation (8) in Geppert (1995)] can}

be used to obtain the optimal hedge ratio for any hedging horizon length once the required parame-ters are estimated. We will describe the methodology proposed by Geppert (1995) later on in the article.

3_{In the empirical analysis, we use nearest-to-maturity futures contracts (with a rollover on the ﬁrst}

day of the contract month) and ignore the impact of maturity on the estimate of the hedge ratio. However, Lee, Budnys, and Lin (1987) ﬁnd that the optimal hedge ratio increases as the futures contracts approach maturity.

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commodities considered in this article do not satisfy this condition. The method used in the article, on the other hand, is valid when the prices are unit-root processes, as well as when the prices are stationary processes. Therefore, the empirical results obtained in this article, which are consistent with the results obtained by JG, should complement the results obtained by JG.

It is also important to note that it is possible, in an empirical analy-sis, to ﬁnd one of the futures price and spot price series to be a unit-root series and the other series to be a stationary series. In this case there can-not be a stationary relationship between the two series, which essentially implies that one cannot use one series (e.g., futures prices) to hedge the risk associated with the other series (e.g., spot prices). However, because the futures and spot prices are related through a no-arbitrage condition, such a situation cannot be theoretically acceptable. Occurrences of such a situation in empirical analyses could result from the low power of the conventional unit-root tests. If such situations do occur, then one is advised to use different unit-root tests with more power.5

In this study we discover that, for almost all of the 25 futures con-tracts, the MV hedge ratios are less than 1 (i.e., the naïve hedge ratio) and also that the hedge ratios increase with the length of hedging hori-zon.6 _{Furthermore, it is found that in-sample hedging effectiveness} increases with the length of hedging horizon, which is consistent with existing findings. For most of the 25 futures contracts, the long-run hedge ratios are close to the naïve hedge ratio, also consistent with the results obtained by JG, who analyzes 5 futures contracts (SP500, German mark, Japanese yen, Swiss franc, and Municipal Bond Index). This suggests that, if the hedging horizon is long, then there is no need to estimate the optimal hedge ratio, because the naïve hedge ratio of 1 will be optimal.7 _{This implication is very important, because the naïve} hedge ratio does not require any data collection and estimation.

5_{We would like to thank an anonymous referee for pointing out this situation to us. For detailed}

information on different types of root tests and the issues regarding the low power of the unit-root tests, see Maddala and Kim (1998).

6_{There are eight contracts for which the hedge ratios are greater than one for some of the hedging}

horizons considered. For almost all of these contracts, the hedge ratios decrease toward 1 except for soybean oil and silver.

7_{The optimal long-run hedge ratios are estimated based on the coefﬁcients of level of prices instead}

of the coefficients of changes in prices, which are closely related to coefficients of the error-correction term. Therefore, the long-run hedge ratio does not correspond to any speciﬁc hedging horizon. Instead, we can only say that the longer the hedging horizon is, the more appropriate will be the use of the long-run hedge ratios. Because most of the short-run hedge ratios (for a hedging horizon up to 8 weeks) are less than the long-run hedge ratios and are approaching 1, we can empir-ically say that the long run is longer than 8 weeks.

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The remainder of this article is organized as follows. In the next sec-tion the methodology is discussed. We then present the empirical results. The article concludes in the ﬁnal section.

METHODOLOGY

The basic concept of hedging is to eliminate (or reduce) ﬂuctuations in the value of a spot position by including futures contracts in the port-folio. Specifically, consider a portfolio consisting of C_s units of a long spot position and C_f units of a short futures position. Let S_t and F_t denote the spot and futures prices at the end of period t, respectively. The change in the value of the hedged portfolio over the period ⌬VHis

then given by

⌬VH⫽ Cs⌬St⫺ Cf⌬Ft (1)

where ⌬St⫽ St⫺ St⫺1and ⌬Ft⫽ Ft⫺ Ft⫺1.

The MV hedge ratio is obtained by minimizing the variance of ⌬VH

and is given by

(2) The conventional approach to estimate the MV hedge ratio involves an estimation of the regression of the changes in spot prices on the changes in futures prices with the use of the OLS regression technique. Speciﬁcally, the regression equation can be written as

(3) where the estimate of the MV hedge ratio H is given by the estimate of ␤.

Before we estimate Equation (3), we need to decide on the differ-encing interval or data frequency. For example, if we use weekly data, then the differencing interval is 1 week. In this case, ⌬Stand ⌬Ft

repre-sent weekly spot and futures price changes, respectively. Because in hedging we are concerned with the change in the value of the portfolio from the beginning to the end of the hedging horizon, the differencing interval should be equal to the length of the hedging horizon. For exam-ple, if the length of the hedging horizon is 1 week, then the differencing interval should be 1 week; that is, weekly data should be used.

The issue of appropriate data frequency or differencing interval for a given length of hedging horizon can be discussed in terms of the way that the MV hedge ratio is derived. In order to simplify the discussion,

¢St⫽ a ⫹ b¢Ft⫹ et

H⫽ Cf Cs

⫽ Cov(¢S,¢F) Var(¢F)

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we assume that one period is equal to 1 week and the length of the hedging horizon (hedging period) consists of an integer number of peri-ods (weeks). As discussed earlier, the MV hedge ratio is obtained by min-imizing the variance of the change in the hedged portfolio’s value, where the change in the portfolio’s value is given by the difference between the values of the hedged portfolio at the beginning and end of the hedging period. For example, if the length of the hedging horizon (or hedging period) is k periods (or k weeks), then the objective should be to mini-mize the variance of the k-period price change. Similarly, in the optimal hedge ratio formula given by Equation (2), the price changes (⌬St and

⌬Ft) should involve k-period differencing for a k-period hedging horizon.

Therefore, when using the regression Equation (3) to estimate the optimal hedge ratio, we need to use k-period differencing for a k-period hedging horizon. In other words, the frequency of data used must corre-spond to the length of the hedging horizon. In order to explain the effect of a mismatch between the hedging period and differencing period, sup-pose that we estimate the hedge ratio using one-period differencing where the length of the hedging horizon is four periods. As described above, since the hedging horizon is four periods, the objective is to min-imize the variance of the change in the value of the portfolio over the 4-week period. The change in the portfolio’s value over the 4-week peri-od is given by

(4) where ⌬4St⫽ (1 ⫺ L4)St⫽ St⫺ St⫺4and ⌬4Ft⫽ (1 ⫺ L4)Ft⫽ Ft⫺ Ft⫺4. In this case the optimal hedge ratio is given by [see Equation (2)]

(5) which can be obtained by estimating regression Equation (3) with the use of four-period (4-week) differencing. However, if one-period (1-week) differencing is used in the regression, then the regression will provide the estimate for the following parameter:

(6) instead of the hedge ratio given by Equation (5). Note that Equation (6) is the optimal hedge ratio if the hedging horizon is one period.

In general, the two hedge ratios estimated by Equations (5) and (6) will not be the same. However, the conditions under which the two

a⫽ Cov(¢St,¢Ft) Var(¢Ft) ⫽ Cov(St ⫺ St⫺1,Ft ⫺ Ft⫺1) Var(Ft⫺ Ft⫺1) H⫽ Cf Cs ⫽ Cov(¢4St,¢4Ft) Var(¢4Ft) ⫽ Cov(St ⫺ St⫺4, Ft⫺ Ft⫺4) Var(Ft⫺ Ft⫺4) ¢4VH⫽ Cs¢4St⫹ Cf¢4Ft

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hedge ratios will be the same can be established. The four-period price changes can be expressed in terms of one-period price changes as follows:

⌬4St⫽ ⌬St⫹ ⌬St⫺1⫹ ⌬St⫺2⫹ ⌬St⫺3 and

⌬4Ft⫽ ⌬Ft⫹ ⌬Ft⫺1⫹ ⌬Ft⫺2⫹ ⌬Ft⫺3

Therefore, if the one-period price changes are serially independent and stationary, then we have

Cov(⌬4St, ⌬4Ft) ⫽ 4 Cov(⌬St, ⌬Ft) and Var(⌬4Ft) ⫽ 4 Var(⌬Ft) (7)

In this case, the two hedge ratios will be the same.8_{However, if the} con-ditions of serial independence and stationarity are not satisﬁed, then the two hedge ratios will be different. Therefore, in general it is inappropri-ate to use a one-period price change (1-week differencing) to estiminappropri-ate the optimal hedge ratio for a four-period hedging horizon.

The regression method used by JG is consistent with our discussion of the relationship between the length of hedging period and data fre-quency used in the estimation of the optimal hedge ratio. For example, JG uses k-period differencing for a k-period hedging horizon in estimat-ing the regression-based MV hedge ratio. Because JG uses approximately 13 months of data for estimating the hedge ratio, he employs overlapping differencing in order to minimize the reduction in sample size caused by multiperiod differencing. However, this will lead to correlated observa-tions, instead of independent observaobserva-tions, and requires the use of regression with autocorrelated errors in the estimation of the hedge ratio. Because our sample size is quite large (with a minimum sample size of 1783 and a maximum sample size of 4956), we use nonoverlapping dif-ferencing in the article.

Instead of using overlapping differencing, we can solve the problem of reduction in the sample size associated with a longer hedging horizon by assuming a specific data-generating process where the hedge ratio for various hedging horizons can be expressed in terms of a few parameters that can easily be estimated. One such technique is suggested by JG and we briefly describe his method. Suppose that the spot and futures prices are both unit-root processes. The market efficiency then implies that the two series would be co-integrated. In this case, the futures and spot

8_{This result can be extended to show that, in general, if the price changes are serially independent}

and stationary, then the frequency of data is irrelevant. In other words, if these conditions hold, then the frequency of data does not need to match the length of the hedging horizon.

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prices can be described by the following processes (see Hylleberg & Mizon, 1989; Stock & Watson, 1988):

(8a) (8b) (8c) (8d) where Pt and ␶tare permanent and transitory factors that drive the spot

and futures prices, and wtand are white-noise processes. Note that Pt

follows a pure random walk process and follows a stationary process. The MV hedge ratio for a k-period hedging horizon is hence given by [see Geppert (1995)].

(9) Based on Equation (9), the long-run hedge ratio (the hedge ratio is where the length of hedging horizon goes to inﬁnity) is given by A1兾B1. It is important to note that the long-run hedge ratio will be equal to the naïve hedge ratio if A1⫽ B1. Therefore, whether the long-run hedge ratio is equal to the naïve hedge ratio is an empirical question. One of the advantages of using Equation (9), instead of a regression, is that it avoids the problem of reduction in the sample size associated with non-overlapping differencing.

It is important to note that the derivation of the hedge ratio given by Equation (9) makes some assumptions. First, the spot and futures prices must be co-integrated, which also require that both prices be unit-root processes. Second, the model implicitly assumes that both the spot and futures price changes have zero expected value, which may not be true in some cases, especially in the case of stock indices. Finally, such a model-based hedge ratio provides a good estimate of the hedge ratio so long as the model [system of Equations (8)] represents the true underlying data-generating process.

In this article we re-examine the regression Equation (3) and modify the equation so that the estimation can be improved. Note that regres-sion Equation (3) represents a relationship between the changes in the series (⌬Stand ⌬Ft), rather than the series themselves (Stand Ft). Let us

consider the following relationship between the series themselves: (10) S_t⫽ a ⫹ bF_t⫹ u_t H⫽ A1B1ksw 2 _{⫹ 2A} 2B2[(1⫺ ak)兾(1 ⫺ a2)]s2n B12ks2w⫹ 2B22[(1⫺ ak)兾(1 ⫺ a2)]s2n t_t n_t t_t⫽ a1tt⫺1⫹ nt,

0ⱕ 冟a1冟 ⬍ 1 Pt⫽ Pt⫺1⫹ wt Ft⫽ B1Pt⫹ B2tt St ⫽ A1Pt⫹ A2tt

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As discussed in the Appendix, Equation (3) represents a short-run rela-tionship between the spot and futures prices, whereas Equation (10) represents a long-run relationship. Therefore, we can estimate Equa-tion (3) to obtain the short-run hedge ratio and EquaEqua-tion (10) to obtain the long-run hedge ratio.9_{Rather than running two separate regressions} to estimate the short- and long-run hedge ratios, we suggest a joint esti-mation of the two hedge ratios using the following regression:

(11) It is important to note that Equation (11) is based on the simultane-ous equation models considered by Hsiao (1997) and Pesaran (1997).10 Furthermore, it is different from the two-step method used by Chou, Fan, and Lee (1996) in the sense that the estimation of Equation (11) is a single-step process. From Equation (11), the long-run hedge ratio is given by and the short-run hedge ratio is given by . The long-run hedge ratio given by corresponds to the long-run hedge ratio of A₁兾B₁ derived by JG. We expect the long-run hedge ratio to remain constant and the short-run hedge ratio to change and approach the long-run hedge ratio as we increase the length of hedging period. It would be interesting to see if the long-run hedge ratio is equal to the naïve hedge ratio of unity. It is important to note that Equation (11) is valid if both the spot and futures price series are stationary. It is also valid if both the series are unit-root processes and are co-integrated.

EMPIRICAL ANALYSIS

This article analyzes 25 different futures contracts where the futures prices are associated with nearest-to-maturity contracts. A list of the futures contracts, sample periods, and sample sizes are given in Table I. The data are obtained from Datastream. The futures contracts used in the study are nearest-to-maturity contracts. The futures contract is rolled over to the next contract on the ﬁrst day of the contract month. In order to see the impact of the length of hedging horizon, various data

⫺a3兾a2

b ⫺a3兾a2

¢St ⫽ a1⫹ a2St⫺1⫹ a3Ft⫺1⫹ b¢Ft⫹ et

9_{Note that there are different short-run hedge ratios associated with different lengths of hedging}

horizons. Different short-run hedge ratios can be estimated with the use of data frequency that matches the length of hedging horizon.

10_{Equation (11) is a version of Equation (8) in Pesaran (1997) and of Equation (2.7) in Hsiao}

(1997). It is parametrized so as to be closely associated with the error-correction models (ECM) encountered in the vector autoregressive (VAR) models with co-integration. However, it is important

to note that Equation (11) is different from the ECM models in that ⌬Ft, instead of ⌬Ft⫺1, appears

on the right-hand side. Furthermore, in ECM there are two equations like Equation (11) (one with

⌬St as the left-hand-side variable and the other with ⌬Ftas the left-hand-side variable), whereas here

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frequencies (ranging from daily to 8 week) are examined. The empirical results are presented below.

In-Sample Analysis

Because the method suggested by JG is applicable only if the futures and spot prices consist of a unit root, it would be interesting to see how many of the 25 commodities satisfy the unit-root condition. In order to simplify computation, we perform the Phillips–Perron (1988) unit-root test with weekly data at the 5% significance level. The results are summarized in Table II. It is important to note that for some contracts (such as Wheat, Coffee, and British pound), both the futures price and spot price series may not be significant at the 5% level. We consider the unit-root condi-tion to be violated if either (a) the t tests for both series are negative and significant at the 5% level or (b) the t test for one of the series is neg-ative and significant at the 5% level and the t test for the other series is

TABLE I

Summary of 25 Futures Contracts

Commodity Sample Period Sample Size

1 SP500 June 1, 1982–December 31, 1997 4066

2 TSE35 March 1, 1991–December 31, 1997 1783

3 Nikkei 225 September 5, 1988–December 31, 1997 2432

4 TOPIX September 5, 1988–December 31, 1997 2432

5 FTSE100 May 3, 1984–December 31, 1997 3564

6 CAC40 March 1, 1989–December 31, 1997 2305

7 All ordinary January 3, 1984–December 31, 1997 3651

8 Soybean oil January 2, 1979–December 31, 1997 4956

9 Soybean January 2, 1979–December 31, 1997 4956

10 Soy meal January 2, 1979–December 31, 1997 4956

11 Corn January 2, 1979–December 31, 1997 4956

12 Wheat March 30, 1982–December 31, 1997 4111

13 Cotton January 3, 1980–December 31, 1997 4694

14 Cocoa November 1, 1983–December 31, 1997 3696

15 Coffee January 2, 1979–December 31, 1997 4956

16 Pork belly January 2, 1979–December 31, 1997 4956

17 Hogs March 30, 1982–December 31, 1997 4111

18 Crude oil April 4, 1983–December 31, 1997 3847

19 Silver January 2, 1979–December 31, 1997 4956

20 Gold January 2, 1979–December 31, 1997 4956

21 Japanese yen January 2, 1986–December 31, 1997 3129

22 Deutsche mark January 2, 1986–December 31, 1997 3129

23 Swiss franc January 2, 1986–December 31, 1997 3129

24 British pound January 2, 1986–December 31, 1997 3129

25 Canadian dollar November 30, 1987–December 31, 1997 2632

Note. This table lists the commodities, sample periods, and sample sizes for the 25 different futures contracts used for empirical analyses in this study. The data are obtained from Datastream.

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negative and signiﬁcant at the 10% level.11_{Based on this criterion, 14 out} of 25 commodities do not satisfy the unit-root condition.12_{Therefore, we} cannot apply JG’s method for these commodities. This shows that JG’s unit-root model is not universally acceptable. However, the method used in this article will be applicable to all the commodities considered in the article.

The results for the OLS estimates of the MV hedge ratios obtained from Equation (3) for various data frequencies are shown in Table III. The hedge ratio changes as the length of return period changes. For

TABLE II

Results of the Unit-Root Tests on Futures and Spot Prices

Phillips–Perron t Test Commodity Sample Size Futures Price Spot Price

SP500 785 2.945** 3.024** TSE35 343 1.013 0.957 Nikkei 225 486 ⫺1.119 ⫺1.141 TOPIX 486 ⫺1.158 ⫺1.172 FTSE100 713 0.819 0.974 CAC40 461 ⫺0.841 ⫺0.714 All ordinary 730 ⫺1.195 ⫺1.138 Soybean oil 991 ⫺3.245** ⫺3.106** Soybean 991 ⫺3.412** ⫺3.422** Soy meal 991 ⫺3.178** ⫺3.088** Corn 991 ⫺3.101** ⫺3.123** Wheat 822 ⫺2.730* ⫺3.272** Cotton 939 ⫺3.540** ⫺3.379** Cocoa 739 ⫺1.592 ⫺1.688 Coffee 991 ⫺3.033** ⫺2.687* Pork belly 991 ⫺3.546** ⫺4.377** Hogs 822 ⫺4.114** ⫺3.985** Crude oil 769 ⫺3.115** ⫺3.140** Silver 991 ⫺2.928** ⫺3.051** Gold 991 ⫺3.326** ⫺3.309** Japanese yen 626 ⫺2.276 ⫺2.285 Deutsche mark 626 ⫺2.917** ⫺2.911** Swiss franc 626 ⫺2.838* ⫺2.851* British pound 626 ⫺2.909** ⫺2.837* Canadian dollar 526 ⫺0.403 ⫺0.306

Note. This table lists the results of the Phillips–Perron unit-root tests on the futures and spot prices based on the weekly data. The critical values at the 10% and 5% signiﬁcance levels are ⫺2.57 and ⫺2.87, respectively. Double asterisks and asterisks represent 5% and 10% signiﬁcance levels, respectively.

11_{This criterion is justiﬁed on the ground that, as explained in the introduction section, the futures}

and spot prices are expected to be either both stationary or both unit root and co-integrated.

12_{Note that the t test for the S&P 500 is signiﬁcantly positive at the 5% level, indicating the}

possi-bility of a second unit root. We have performed the unit-root test on the first differenced series and ﬁnd them to be stationary. This conﬁrms that the S&P 500 futures and spot prices have a single unit root.

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T

ABLE III

The OLS Estimates of the Minimum-V

ariance Hedge Ratios for Different T

ypes of Futures Contracts

Holding P eriod Commodity 1 Day 1 W eek 2 W eek 3 W eek 4 W eek 5 W eek 6 W eek 7 W eek 8 W eek SP500 Hedge ratio 0.8314 0.8978 0.9489 0.9421 0.9880 0.9372 0.971 1 0.9754 0.9950 Adj. R 2 0.9176 0.9546 0.9716 0.9793 0.9792 0.9785 0.9850 0.9839 0.9915 TSE35 Hedge ratio 0.8202 0.9463 0.9636 0.9755 0.9744 0.9674 0.9894 1.0200 0.9910 Adj. R 2 0.7607 0.9309 0.9385 0.9436 0.9341 0.9726 0.9488 0.9699 0.9821 Nikkei 225 Hedge ratio 0.9198 0.9840 0.9852 0.9593 0.9799 0.9562 0.9644 0.9621 0.971 1 Adj. R 2 0.8374 0.9410 0.9627 0.9703 0.9751 0.9753 0.9814 0.9926 0.9799 T OPIX Hedge ratio 0.7761 0.9233 0.9565 0.9479 0.9563 0.9536 0.9473 0.9660 0.9898 Adj. R 2 0.7587 0.9228 0.9543 0.9600 0.9746 0.9705 0.9783 0.9860 0.9831 FTSE100 Hedge ratio 0.7857 0.8650 0.8995 0.9148 0.9230 0.9329 0.9415 0.9175 0.9156 Adj. R 2 0.8814 0.9520 0.9630 0.9642 0.9658 0.9707 0.9745 0.9751 0.9792 CAC40 Hedge ratio 0.8843 0.9405 0.9581 0.9538 0.9704 0.9518 0.9473 0.9510 0.9749 Adj. R 2 0.9186 0.9662 0.9765 0.9769 0.9839 0.9825 0.9806 0.9812 0.9827 All ordinary Hedge ratio 0.2970 0.7744 0.8254 0.8300 0.8687 0.8783 0.8527 0.9605 0.8341 Adj. R 2 0.1708 0.8307 0.9020 0.9238 0.9443 0.9504 0.9509 0.9597 0.9443 Soybean oil Hedge ratio 0.8741 0.9324 0.9593 0.9690 0.9990 1.0101 0.9962 1.0344 1.0502 Adj. R 2 0.6528 0.8329 0.8672 0.8801 0.91 18 0.9322 0.9182 0.9451 0.9418 Soybean Hedge ratio 0.8677 0.9062 0.9089 0.8561 0.8856 0.9031 0.91 13 0.9153 0.9208 Adj. R 2 0.7445 0.8518 0.8566 0.8528 0.8880 0.8995 0.8993 0.91 10 0.9269 Soy meal Hedge ratio 0.8435 0.9052 0.9680 0.9704 0.9364 0.9572 0.9357 0.9535 0.9004 Adj. R 2 0.6559 0.7860 0.8178 0.8074 0.7757 0.7978 0.8606 0.8488 0.8425 Corn Hedge ratio 0.6873 0.8047 0.8374 0.7764 0.8397 0.8683 0.7999 0.8598 0.8798 Adj. R 2 0.4865 0.6097 0.6040 0.6663 0.6817 0.6944 0.6617 0.7470 0.7109 Wheat Hedge ratio 0.7200 0.8459 0.9264 0.8727 0.9661 0.7723 0.9819 0.7860 1.0625 Adj. R 2 0.3907 0.6022 0.6936 0.6679 0.6561 0.6270 0.7199 0.6261 0.7235 Cotton Hedge ratio 0.4720 0.51 17 0.5352 0.5807 0.5916 0.5676 0.5681 0.5728 0.8537 Adj. R 2 0.2669 0.3160 0.3602 0.3943 0.4189 0.3848 0.3790 0.4146 0.7905

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Cocoa Hedge ratio 0.8472 0.9301 0.9284 0.9176 0.9725 0.9036 0.9446 0.921 1 Adj. R 2 0.4975 0.7929 0.8028 0.7980 0.8263 0.7799 0.7978 0.8254 Cof fee Hedge Ratio 0.51 14 0.6301 0.7242 0.8215 0.7568 0.8152 0.9025 0.7856 Adj. R 2 0.3303 0.4903 0.5555 0.7700 0.5618 0.7597 0.8833 0.7294 Pork belly Hedge Ratio 0.3743 0.7222 0.7481 0.6839 0.7966 0.7957 0.6571 0.7063 Adj. R 2 0.0762 0.2477 0.3227 0.3203 0.4800 0.4538 0.3158 0.3973 Hogs Hedge Ratio 0.1253 0.2634 0.321 1 0.4120 0.4103 0.4423 0.51 1 1 0.4178 Adj. R 2 0.0204 0.1349 0.2107 0.2869 0.2678 0.3548 0.4108 0.3307 Crude oil Hedge Ratio 0.8362 0.9435 0.9816 1.0047 0.9973 1.0104 1.0294 0.9912 Adj. R 2 0.5693 0.8588 0.9277 0.9422 0.9527 0.9692 0.9812 0.9856 Silver Hedge Ratio 0.4732 0.7725 0.9204 1.0856 0.8849 0.9499 1.1706 0.9151 Adj. R 2 0.1 173 0.4144 0.5996 0.6724 0.8273 0.9255 0.9363 0.7173 Gold Hedge Ratio 0.4980 0.8662 0.9275 1.0146 0.9714 0.8586 0.9814 0.9574 Adj. R 2 0.2044 0.7943 0.8854 0.9512 0.9530 0.9513 0.9702 0.9825 Japanese Hedge Ratio 0.9179 0.9575 0.9797 0.9582 0.9704 0.9830 0.9844 0.9705 yen Adj. R 2 0.8781 0.9591 0.9686 0.9725 0.9834 0.9849 0.9830 0.9870 Deutsche Hedge Ratio 0.9317 0.9792 0.9922 0.9957 0.9943 0.9906 1.0096 0.9897 mark Adj. R 2 0.8950 0.9687 0.9798 0.9827 0.9864 0.9851 0.9879 0.9913 Swiss franc Hedge Ratio 0.9286 0.9757 0.9880 0.9841 0.9848 0.9882 0.9989 0.9846 Adj. R 2 0.8972 0.9734 0.9797 0.9835 0.9893 0.9863 0.9904 0.9924 British pound Hedge Ratio 0.8863 0.9240 0.9322 0.9441 0.9800 0.9673 0.9627 0.9783 Adj. R 2 0.8531 0.9176 0.9394 0.9468 0.9826 0.9783 0.9609 0.9851 Canadian Hedge Ratio 0.8293 0.9085 0.9323 0.9558 0.9391 0.9692 0.9272 0.9241 dollar Adj. R 2 0.8005 0.8919 0.9188 0.9273 0.9395 0.9652 0.9451 0.9564 Note. This table presents the results for the OLS estimates of the minimum-variance hedge ratios obtained from Equation (3) for vario us holding-period returns for each of the contracts listed in T able I.

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most stock index futures, the hedge ratio seems to approach the naïve hedge ratio of unity as the hedge period lengthens. However, for some other commodities (e.g., soy meal, hogs, and gold), there does not seem to be any trend towards the naïve hedge ratio. If we combine all the hedge ratios for all of the 25 contracts and 9 hedging horizons, then the mean hedge ratio equals 0.8748, with a standard deviation of 0.1649. This indicates that the short-run hedge ratio is signiﬁcantly less than the naïve hedge ratio (with a t ratio equal to ⫺11.385).

In order to test for the impact of the length of hedging horizon on the hedge ratio as well as hedging effectiveness, the following regres-sions are estimated using the hedge ratios and R2_{obtained from the} esti-mation of regression Equation (3) for all 25 contracts and all 9 hedging horizons:13

(12) (13) where Ti represents the length of hedging horizon expressed in weeks.

The results of the regression are presented in Panel A of Table IV. Because both the estimates of and are highly signiﬁcant and posi-tive, this implies that both the hedge ratio and hedging effectiveness increase with the length of hedging horizon.

The regression Equations (12) and (13), however, do not capture the long-run trend in the sense that positive values of and imply that the hedge ratio as well as hedging effectiveness do not converge to finite values as the length of hedging horizon approaches infinity. Therefore, the following two regressions are also estimated with the use of the same estimated hedge ratios and R2:

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(15) The results of the regression are presented in Panel B of Table IV. The long-run beta is obtained by letting Ti approach infinity in

Equation (14), and thus the long-run beta (hedge ratio) is estimated by

R2 i ⫽ ␼0⫹ ␼1e␼2Ti 1⫹ e␼2Ti ⫹ error b_i⫽ m0⫹ m₁em2Ti 1⫹ em2Ti ⫹ error g₁ f₁ g₁ f₁ Ri2⫽ g0⫹ g1Ti⫹ ni b_i ⫽ f0⫹ f1Ti⫹ ei

13_{It is important to note that, for Equations (12) and (13) to be valid, and later for Equations (14) and}

(15), too, we need to assume a homogeneous behavior across the commodities with respect to the hedging effectiveness and hedge ratio. Such a strong assumption is required in order to increase the sample size by combining all the commodities.

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. The estimate of is equal to 0.9046, and the standard error of is estimated as 0.0127. Therefore, the long-run hedge ratio seems to be signiﬁcantly less than the naïve hedge ratio. However, we will present a better way of estimating the long-run hedge ratio later.

In much the same way, we can estimate the long-run effectiveness of the hedge ratio using the estimate of . The estimate of is equal to 0.8486, and its standard error is equal to 0.0187. This provides some evidence that, even in the long run, the effectiveness of the long-run hedge ratio will not approach one. This result is different from the analytical result shown by JG, where the degree of hedging effectiveness approaches unity as the hedging horizon approaches inﬁnity. The difference in results could be due to the fact that the major-ity of the commodities considered here do not satisfy the unit-root con-dition assumed by JG.

It is important to note that Equation (15) is bounded below by and bounded above by . Therefore, if the hedging horizon approaches 0 (instantaneous hedging), then the hedging

␼0⫹ ␼1 ␼0⫹ 0.5␼1 ␼0⫹ ␼1 ␼0⫹ ␼1 m₀⫹ m1 m₀⫹ m1 m₀⫹ m1 TABLE IV

Effect of Hedging Horizon Length on the Hedge Ratio and Hedging Effectiveness

Panel A: Equations (12) and (13)

Equation (12) ␾0 ␾1 Adjusted R2 0.8032 0.0178 0.0760 (40.58)(4.28) Equation (13) ␥0 ␥1 Adjusted R2 0.6957 0.0268 0.0852 (24.89)(4.56)

Panel B: Equations (14) and (15)

Equation (14) ␮0 ␮1 ␮2 Adjusted R2 0.4552 0.4494 1.7551 0.1195 (5.05)(4.93)(2.67) Equation (15) ␼0 ␼1 ␼2 Adjusted R2 0.2379 0.6107 1.5042 0.1188 (1.99)(5.03)(2.71)

Note. This table presents the results for the impact of hedging horizon’s length on the hedge ratio and hedging effectiveness. Panel A presents the results for regression Equations (12) and (13), and Panel B presents the results for regression Equations (14) and (15). The t values are in parentheses.

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effectiveness will be 54.33% ( ). However, Equation (15) does not impose the restriction that the function is bounded by 0 and 1. Fortunately, in our empirical analysis, the estimates of the upper bound and the lower bound lie between 0 and 1. If such a condition is violated in the empirical analysis, one needs to look for another functional form that satisﬁes such a restriction.14_{We choose the} function, because it is an improvement over Equation (13).

The empirical results so far are based on the estimation of Equa-tion (3). As menEqua-tioned earlier, it is better to estimate EquaEqua-tion (11), instead of Equation (3), in order to estimate the optimal hedge ratio. If Equation (11) is a better speciﬁcation compared to Equation (3), then the adjusted R2_{associated with Equation (11) should be higher} com-pared to the one associated with Equation (3). The results of the estima-tion of Equaestima-tion (11) are presented in Table V. Out of 225 adjusted R2 (25 commodities with 9 hedging horizons), 211 adjusted R2 _reported in Table V are higher than the corresponding ones reported in Table III. The average adjusted R2 _{associated with Equation (11) is 84.88 percent,} whereas the average adjusted R2 _{associated with Equation (3) is 80.33%.} This clearly indicates that Equation (11) is a better way of estimating the short-run hedge ratio.

It is also important to note that, for each futures contract, the short-run hedge ratios are different for different hedging horizons. However, the estimates of the long-run hedge ratios should be close to each other, because we are estimating the same long-run hedge ratio regardless of the data frequencies (differencing period) being used. The long-run hedge ratios from Table V seem to be close to 1, with the average equal to 1.0073 and a standard deviation equal to 0.0608. Therefore, the esti-mates of the long-run hedge ratios are not signiﬁcantly different from the naïve hedge ratio. This result on the long-run hedge ratio is consis-tent with the empirical result obtained by JG, indicating that if the hedg-ing horizon is long, then the naïve technique can be quite effective. As for the short-run hedge ratios estimated from Equation (11) and reported in Table V, they are found to be signiﬁcantly less than 1, with the average hedge ratio of 0.8875 and a standard deviation of 0.1618, which results in a t ratio of ⫺11.433.

Even though the short-run hedge ratios are found to be signifi-cantly less than 1, there are some commodities for which the estimated

␼0⫹ 0.5␼1⫽ 0.5433

14_{This issue is similar to the one we face when estimating the variance or standard deviation}

param-eter. It is better if we get the estimate of the variance parameter to be positive without having to impose the restriction in the estimation procedure that will guarantee the estimate to be positive.

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T

ABLE V

Simultaneous Estimation of the Short- and Long-Run Hedge Ratios for Different T

ypes of Futures Contracts

Holding P eriod Commodity 1 Day 1 W eek 2 W eek 3 W eek 4 W eek 5 W eek 6 W eek 7 W eek 8 W SP500 ␤ 0.839 0.908 0.952 0.947 0.983 0.954 0.973 0.969 0.978 (0.004) (0.006)(0.007)(0.007)(0.008)(0.009)(0.008)(0.009)(0.008) ⫺ ␣3 兾␣ 2 0.995 0.995 0.995 0.996 0.995 0.997 0.996 0.996 0.995 Adj. R 2 0.928 0.965 0.979 0.987 0.987 0.988 0.992 0.991 0.995 TSE35 ␤ 0.841 0.961 0.983 0.996 1.002 0.988 1.000 0.993 0.967 (0.010) (0.012)(0.015)(0.016)(0.018)(0.014)(0.021)(0.019)(0.012) ⫺ ␣3 兾␣ 2 1.003 1.002 1.002 1.003 1.000 1.001 1.003 1.005 1.004 Adj. R 2 0.799 0.949 0.963 0.970 0.972 0.987 0.976 0.986 0.995 Nikkei 225 ␤ 0.926 0.986 0.988 0.972 0.991 0.974 0.974 0.967 1.003 (0.008) (0.010)(0.01 1) (0.01 1) (0.01 1) (0.012)(0.012)(0.009)(0.014) ⫺ ␣3 兾␣ 2 0.979 0.977 0.977 0.977 0.975 0.978 0.975 0.979 0.977 Adj. R 2 0.856 0.954 0.972 0.981 0.985 0.986 0.988 0.994 0.989 T OPIX ␤ 0.793 0.937 0.965 0.967 0.968 0.968 0.964 0.970 1.002 (0.008) (0.01 1) (0.012)(0.013)(0.012)(0.013)(0.013)(0.012)(0.014) ⫺ ␣3 兾␣ 2 0.978 0.976 0.976 0.976 0.975 0.977 0.973 0.979 0.976 Adj. R 2 0.791 0.940 0.965 0.974 0.983 0.983 0.986 0.990 0.989 FTSE100 ␤ 0.789 0.874 0.91 1 0.927 0.936 0.956 0.949 0.937 0.937 (0.005) (0.007)(0.009)(0.010)(0.01 1) (0.012)(0.012)(0.012)(0.013) ⫺ ␣3 兾␣ 2 0.999 1.000 0.999 0.999 1.000 0.999 0.997 1.001 1.003 Adj. R 2 0.890 0.958 0.969 0.972 0.975 0.980 0.981 0.984 0.985 CAC40 ␤ 0.888 0.945 0.962 0.959 0.978 0.964 0.963 0.969 0.995 (0.005) (0.008)(0.009)(0.01 1) (0.01 1) (0.012)(0.013)(0.014)(0.015) ⫺ ␣3 兾␣ 2 1.010 1.010 1.012 1.013 1.010 1.01 1 1.014 1.019 1.004 Adj. R 2 0.923 0.969 0.980 0.981 0.987 0.986 0.986 0.988 0.989 (Continued

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T

ABLE V

ypes of Futures Contracts (

Continued ) Holding P eriod Commodity 1 Day 1 W eek 2 W eek 3 W eek 4 W eek 5 W eek 6 W eek 7 W eek 8 W eek All ordinary ␤ 0.320 0.784 0.822 0.836 0.865 0.883 0.872 0.964 0.877 (0.009) (0.01 1) (0.012)(0.012)(0.012)(0.013)(0.014)(0.014)(0.015) ⫺ ␣3 兾␣ 2 0.987 0.987 0.989 0.989 0.988 0.988 0.992 0.990 0.988 Adj. R 2 0.427 0.875 0.933 0.951 0.966 0.972 0.969 0.980 0.974 Soybean oil ␤ 0.876 0.936 0.965 0.976 1.004 1.016 1.005 1.040 1.053 (0.009) (0.013)(0.017)(0.019)(0.020)(0.019)(0.023)(0.021)(0.023) ⫺ ␣3 兾␣ 2 1.092 1.098 1.094 1.102 1.1 14 1.099 1.106 1.1 1 1 1.122 Adj. R 2 0.658 0.839 0.874 0.889 0.918 0.938 0.928 0.951 0.948 Soybean ␤ 0.868 0.900 0.894 0.845 0.870 0.891 0.891 0.897 0.896 (0.007) (0.012)(0.016)(0.017)(0.018)(0.020)(0.021)(0.021)(0.021) ⫺ ␣3 兾␣ 2 0.968 0.979 0.989 0.980 0.987 0.983 0.994 0.984 0.986 Adj. R 2 0.752 0.865 0.877 0.888 0.915 0.922 0.929 0.939 0.948 Soy meal ␤ 0.843 0.906 0.966 0.967 0.936 0.960 0.940 0.955 0.916 (0.009) (0.015)(0.020)(0.025)(0.031)(0.033)(0.029)(0.033)(0.036) ⫺ ␣3 兾␣ 2 1.041 1.048 1.051 1.047 1.041 1.026 1.038 1.032 1.150 Adj. R 2 0.660 0.793 0.827 0.823 0.799 0.821 0.876 0.868 0.849 Corn ␤ 0.688 0.799 0.806 0.758 0.815 0.831 0.782 0.838 0.823 (0.010) (0.020)(0.029)(0.027)(0.032)(0.035)(0.034)(0.033)(0.036) ⫺ ␣3 兾␣ 2 1.070 1.083 1.082 1.077 1.067 1.092 1.067 1.073 1.076 Adj. R 2 0.496 0.636 0.664 0.745 0.769 0.787 0.817 0.856 0.866 Wheat ␤ 0.723 0.844 0.914 0.854 0.912 0.766 0.913 0.754 0.926 (0.014) (0.024)(0.030)(0.036)(0.047)(0.043)(0.050)(0.050)(0.064) ⫺ ␣3 兾␣ 2 0.932 0.946 0.973 0.960 0.982 0.953 0.992 0.942 0.993 Adj. R 2 0.401 0.622 0.719 0.710 0.71 1 0.707 0.777 0.731 0.796 Cotton ␤ 0.474 0.517 0.547 0.589 0.610 0.585 0.596 0.621 0.869 (0.01 1) (0.023)(0.029)(0.036)(0.037)(0.040)(0.044)(0.045)(0.034) ⫺ ␣3 兾␣ 2 1.043 1.047 1.052 1.047 1.045 1.063 1.044 1.046 1.074 Adj. R 2 0.284 0.389 0.502 0.547 0.647 0.665 0.674 0.696 0.870

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Cocoa ␤ 0.849 0.932 0.931 0.922 0.975 0.91 1 0.953 0.924 0.975 (0.014) (0.017)(0.024)(0.029)(0.033)(0.039)(0.043)(0.041)(0.052) ⫺ ␣3 兾␣ 2 0.998 0.994 0.992 1.001 1.001 1.01 1 0.982 1.017 1.003 Adj. R 2 0.503 0.795 0.807 0.804 0.832 0.790 0.808 0.833 0.805 Cof fee ␤ 0.513 0.638 0.727 0.839 0.780 0.858 0.926 0.824 0.937 (0.010) (0.020)(0.027)(0.024)(0.038)(0.031)(0.025)(0.038)(0.041) ⫺ ␣3 兾␣ 2 1.145 1.147 1.151 1.152 1.141 1.132 1.154 1.156 1.173 Adj. R 2 0.349 0.535 0.630 0.798 0.673 0.805 0.899 0.784 0.821 Pork belly ␤ 0.385 0.764 0.804 0.754 0.851 0.866 0.805 0.831 0.884 (0.018) (0.038)(0.046)(0.050)(0.048)(0.055)(0.064)(0.063)(0.057) ⫺ ␣3 兾␣ 2 1.059 1.017 1.035 1.032 1.063 1.051 1.051 1.044 1.098 Adj. R 2 0.102 0.330 0.435 0.465 0.605 0.615 0.570 0.619 0.726 Hogs ␤ 0.125 0.270 0.329 0.421 0.415 0.465 0.513 0.463 0.459 (0.013) (0.022)(0.028)(0.035)(0.042)(0.041)(0.045)(0.046)(0.048) ⫺ ␣3 兾␣ 2 0.803 0.826 0.836 0.812 0.809 0.810 0.814 0.798 0.806 Adj. R 2 0.059 0.229 0.360 0.462 0.485 0.562 0.637 0.612 0.626 Crude oil ␤ 0.862 0.969 1.006 1.008 1.004 0.997 1.017 0.996 0.996 (0.010) (0.010)(0.01 1) (0.01 1) (0.013)(0.01 1) (0.010)(0.008)(0.01 ⫺ ␣3 兾␣ 2 1.005 1.005 1.006 1.006 1.003 1.006 1.006 0.998 1.006 Adj. R 2 0.703 0.929 0.959 0.971 0.970 0.985 0.989 0.993 0.990 Silver ␤ 0.617 1.024 0.969 1.262 0.879 0.979 1.155 1.180 1.052 (0.018) (0.025)(0.021)(0.027)(0.017)(0.013)(0.014)(0.029)(0.009) ⫺ ␣3 兾␣ 2 0.986 0.965 0.987 0.980 0.954 0.953 1.041 0.909 0.965 Adj. R 2 0.214 0.634 0.848 0.875 0.931 0.968 0.980 0.930 0.992 Gold ␤ 0.572 0.889 0.933 0.993 0.947 0.888 0.976 0.959 0.949 (0.010) (0.01 1) (0.01 1) (0.010)(0.01 1) (0.010)(0.010)(0.008)(0.010) ⫺ ␣3 兾␣ 2 0.979 0.968 0.975 0.965 0.978 0.974 0.969 0.960 0.977 Adj. R 2 0.624 0.884 0.938 0.971 0.973 0.977 0.985 0.991 0.987 Japanese yen ␤ 0.922 0.964 0.985 0.972 0.978 0.995 0.994 0.984 0.981 (0.006) (0.008)(0.009)(0.010)(0.009)(0.010)(0.01 1) (0.01 1) (0.009) ⫺ ␣3 兾␣ 2 0.986 0.985 0.985 0.984 0.985 0.985 0.984 0.985 0.986 Adj. R 2 0.887 0.963 0.974 0.978 0.987 0.989 0.989 0.990 0.994 (Continued

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T

ABLE V

ypes of Futures Contracts (

Continued ) Holding P eriod Commodity 1 Day 1 W eek 2 W eek 3 W eek 4 W eek 5 W eek 6 W eek 7 W eek 8 W eek Deutsche mark ␤ 0.934 0.981 0.994 0.997 0.997 0.992 1.010 0.992 1.001 (0.006) (0.007)(0.008)(0.009)(0.009)(0.01 1) (0.01 1) (0.010)(0.010) ⫺ ␣3 兾␣ 2 1.008 1.007 1.006 1.006 1.006 1.006 1.006 1.001 1.003 Adj. R 2 0.899 0.970 0.981 0.984 0.988 0.987 0.989 0.992 0.994 Swiss franc ␤ 0.931 0.977 0.990 0.986 0.987 0.989 0.999 0.987 0.986 (0.005) (0.006)(0.008)(0.008)(0.008)(0.010)(0.009)(0.009)(0.008) ⫺ ␣3 兾␣ 2 0.992 0.991 0.990 0.989 0.988 0.989 0.987 0.982 0.983 Adj. R 2 0.903 0.975 0.982 0.986 0.991 0.988 0.992 0.993 0.995 British pound ␤ 0.894 0.940 0.957 0.974 0.992 0.984 0.998 0.992 1.006 (0.006) (0.010)(0.012)(0.013)(0.009)(0.01 1) (0.016)(0.012)(0.01 1) ⫺ ␣3 兾␣ 2 1.021 1.020 1.022 1.020 1.024 1.023 1.023 1.029 1.018 Adj. R 2 0.865 0.933 0.956 0.965 0.987 0.985 0.977 0.989 0.993 Canadian dollar ␤ 0.839 0.930 0.961 0.989 0.978 0.997 0.969 0.959 0.952 (0.008) (0.013)(0.016)(0.019)(0.019)(0.017)(0.022)(0.021)(0.022) ⫺ ␣3 兾␣ 2 1.039 1.040 1.040 1.040 1.038 1.041 1.039 1.041 1.040 Adj. R 2 0.808 0.904 0.931 0.943 0.953 0.973 0.961 0.968 0.971 Note. This table presents the results for the simultaneous estimation of the short- and long-run hedge ratios obtained from Equation (1 1) with the use of various holding-period returns for each of the futures contracts listed in T able I. The short-run hedge ratios are given under the row heading b , and the long-run hedge ratios are given under the row heading .

Standard errors are reported in parentheses.

⫺

a3

兾

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short-run hedge ratios are greater than 1. They include TSE35 (4 and 6 week), Nikkei 225 (8 week), TOPIX (8 week), Silver (1, 3, 6, 7, and 8 week), Crude Oil (6 week), deutsche mark (6 and 8 week), British pound (8 week), and soybean oil (4, 5, 6, 7, and 8 week). However, all of these hedge ratios move back to within 1% of 1 as the hedging horizon lengthens, except for soybean oil and silver, which do not seem to con-verge to 1. These two commodities are interesting cases and need further analysis, and we intend to do this in the future.

Out-of-Sample Analysis

So far we have discussed the results based on the in-sample analysis. It will be interesting to see the out-of-sample effectiveness of the different hedge ratios as we increase the hedging horizon. Here, we consider the out-of-sample effectiveness of three hedging strategies. The ﬁrst hedging strategy involves the estimation of the short-run hedge ratio using week-ly data, where the hedge ratio is kept the same as the hedging horizon extends from 1 week to 8 weeks. We call this hedge ratio the 1-week hedge ratio. The second strategy involves the estimation of the hedge ratio using the data such that the data frequency matches the hedging horizon. For example, we use i-week differencing to estimate the short-run hedge ratio for an i-week hedging horizon. Such hedge ratios will be referred to as the horizon-adjusted hedge ratio. Finally, the third hedging strategy uses the naïve hedge ratio of unity. We analyze the hedg-ing effectiveness of each of these three hedge ratios, where the hedghedg-ing effectiveness is computed as follows:

(16) In the analysis of the out-of-sample performance, the post sample represents the integer number of years’ worth of data, which approxi-mately covers half of the total sample size.15 The ﬁrst part of the data, which excludes the post sample period, is used to compute the optimal hedge ratios. The out-of-sample hedging effectiveness for the three hedge ratios are summarized in Table VI. The out-of-sample analysis is performed for one equity contract (All Ordinary), one commodity con-tract (Cotton), and one currency concon-tract (Canadian dollar). It would be

Hedging effectiveness ⫽ 1 ⫺ Var(¢VH)

Var(¢S)

15_{For example, for the All Ordinary index, there are 730 weeks’ (14 years and 2 weeks) worth of data.}

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T

A

BLE VI

Out-of-Sample Hedging Effectiveness

Investment Horizon 1 W eek 2 W eek 3 W eek 4 W eek 5 W eek 6 W eek 7 W eek 8 W eek P anel A:

All Ordinary Stock Index

One-week hedge ratio

0.903 0.912 0.922 0.932 0.929 0.936 0.934 0.929

Horizon-adjusted hedge ratio

0.903 0.924 0.943 0.951 0.964 0.966 0.975 0.966

Naïve hedge ratio

0.884 0.889 0.928 0.922 0.957 0.957 0.971 0.969 Relative ef

fectiveness of 1-week hedge ratio

1.000 0.986 0.978 0.980 0.964 0.968 0.958 0.962 Relative ef

fectiveness of horizon-adjusted hedge ratio

1.021 1.040 1.016 1.031 1.007 1.009 1.004 0.997 P

anel B: Cotton _{One-week hedge ratio}

0.459 0.504 0.599 0.572 0.601 0.638 0.627 0.560

0.459 0.513 0.615 0.584 0.616 0.689 0.654 0.669

Naïve hedge ratio

0.215 0.341 0.504 0.506 0.517 0.708 0.659 0.653 Relative ef

1.000 0.982 0.974 0.979 0.975 0.926 0.959 0.837 Relative ef

2.139 1.506 1.220 1.154 1.191 0.973 0.992 1.025 P

anel C: Canadian Dollar _{One-week hedge ratio}

0.924 0.958 0.969 0.967 0.979 0.968 0.979 0.975

0.924 0.959 0.969 0.968 0.976 0.966 0.978 0.980

Naïve hedge ratio

0.918 0.956 0.968 0.967 0.976 0.962 0.972 0.983 Relative ef

1.000 1.000 1.000 0.999 1.003 1.001 1.001 0.995 Relative ef

1.006 1.003 1.001 1.001 1.000 1.004 1.005 0.997 Note. This table presents the out-of-sample hedging ef fectiveness of three dif ferent hedge ratios for three futures contracts. The re lative hedging ef fectiveness of the 1-week hedge ratio measures the ef fectiveness of the 1-week hedge ratio relative to the ef fectiveness of the horizon-adjusted hedge ratio and is c omputed as the ratio of the hedging ef fectiveness of the 1-week hedge ratio to the hedging ef fectiveness of the horizon-adjusted hedge ratio. Similarly , the relative hedging ef fectiven ess of the horizon-adjusted hedge ratio is computed as the

ratio of the hedging ef

fectiveness of the horizon-adjusted hedge ratio to the ef

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interesting to see the performance of the 1-week hedge ratio in relation to the effectiveness of the horizon-adjusted hedge ratio, and, similarly, the relative performance of the horizon-adjusted hedge ratio vis-à-vis the naïve hedge ratio. The relative performance of the 1-week hedge ratio is computed as the ratio of the effectiveness of the 1-week hedge ratio to the effectiveness of the horizon-adjusted hedge ratio. We expect the rel-ative performance of the 1-week hedge ratio to decrease as the hedging horizon lengthens. Similarly, the relative performance of the horizon-adjusted hedge ratio is computed as the ratio of the effectiveness of the horizon-adjusted hedge ratio to the effectiveness of the naïve hedge ratio. Again, we expect the relative performance of the horizon-adjusted hedge ratio to decrease as the hedging horizon is extended. Table VI also summarizes the relative hedging effectiveness for the three contracts. The relative effectiveness is also plotted in Figure 1.

It is clear from Table VI and Figure 1 that the relative performances of both the 1-week and horizon-adjusted hedge ratios decline as the hedging horizon increases. However, the decreasing trend is not as clear for the Canadian dollar as for the other two contracts considered. If we assume the trend to be the same for all three contracts and run a regres-sion of the relative performance on a constant and time trend [similar to Equation (12), where is replaced by the relative hedging perform-ance], then the t statistics, for the coefficients of the time trend, are found to be equal to ⫺2.671 for the relative performance of the 1-week hedge ratio and ⫺2.219 for the relative performance of the horizon-adjusted hedge ratio. This supports our earlier ﬁnding that the long-run hedge ratio is close to the naïve hedge ratio.

CONCLUSIONS

In this article we have estimated the minimum variance hedge ratios for nine different hedging horizons as well as for 25 different commodities. Furthermore, we analyzed the effects of the length of hedging horizon on the optimal hedge ratio and hedging effectiveness. The empirical results indicate that the hedge ratios are signiﬁcantly less than one and increase with the length of hedging horizon. It is also found that hedging effec-tiveness increases with the length of hedging horizon. However, the degree of hedging effectiveness does not approach 1.

We also ﬁnd that for 14 out of 25 different commodities, the unit-root condition is rejected. This implies that the unit-unit-root model used by Geppert (1995) cannot be universally acceptable.

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This article further presents a model that can be used to simultane-ously estimate the short- and long-run hedge ratios. This model is found to be more suitable compared to the conventional regression model. The long-run hedge ratio is found to be close to the naïve hedge ratio of unity.

All Ordinary

Rel. Per. 1 Rel. Per. 2

1 2 3 4 5 1.06 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1.04 1.02 1.00 0.98 1.02 1.00 1.01 0.99 0.98 0.96 0.94 0.8 Cotton 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Canadian Dollar

Hedging Horizon (weeks)

FIGURE 1

Out-of-sample relative hedging effectiveness of the 1-week and horizon-adjusted hedge ratios. “Rel. Per. 1” denotes the relative performance of the 1-week hedge ratio and

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This implies that if the hedging horizon is long, then the naïve hedge ratio, which does not need any estimation, will be close to the minimum variance hedge ratio. This result is similar to that obtained by Geppert (1995), who imposes restrictions like unit-root and co-integration in the data-generating process. However, no such restriction is imposed in the model used in the article, and hence the results obtained in the article complement the results of Geppert (1995). This implies that the equality of the long-run hedge ratio to the naïve hedge ratio seems to be a more general phenomenon. This inference is also supported by the out-of-sample analysis. By contrast, the short-run hedge ratios (estimated using the new model) are found to be signiﬁcantly less than 1.

APPENDIX

Consider two time series y_t and x_t. We can estimate the relationship between the two series using the following two equations:

(A1) and

(A2) where

and

In this appendix we will explain why Equation (A1) can be consid-ered as a long-run relationship and Equation (A2) can be considconsid-ered as a short-run relationship. There are three different possible explanations for the long- and short-run relationships, respectively represented by Equations (A1) and (A2). The ﬁrst explanation is related to the frequency-domain approach, the second one to the distributed-lag model, and the third one to the concept of co-integration.

Frequency-Domain Explanation

In the frequency-domain analysis, it is well known that the differenced series represent the high-frequency component of the original series, which can be shown by using the spectral density of each series. Let

fx(w) and fy(w), respectively, denote the spectral density of the series xt

and yt. The spectral densities of the ﬁrst differenced series ⌬xt and ⌬yt

are then given by

(A3) (A4) f_¢y(w) ⫽ fy(w)g(w) f_¢x(w) ⫽ fx(w)g(w) ¢xt⫽ (1 ⫺ L)xt⫹1⫽ xt⫹1⫺ xt ¢yt ⫽ (1 ⫺ L)yt⫹1⫽ yt⫹1⫺ yt ¢yt⫽ b0⫹ b1¢xt⫹ et yt ⫽ a0⫹ a1xt⫹ ut

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and

(A5) where w represents the frequency. In other words, the spectral density of the differenced series is given by the product of the spectral density of the original series and the function g(w), which represents the differenc-ing operator in the frequency domain. The dependence of function g(w) on w is shown below:

It is important to note that function g(w) approaches zero as the fre-quency approaches zero. This implies that the spectral density of the dif-ferenced series is equal to the spectral density of the original series, where lower-frequency components are eliminated or substantially reduced. Therefore, the differenced series represent the high-frequency component of the original series, and the relationship between the dif-ferenced series [Equation (A2)] represents a high-frequency relation-ship. Because the low-frequency components consist of long-term move-ments and the high-frequency components consist of short-term movements, Equation (A2) can be regarded as a short-run relationship and Equation (A1) can be regarded as a long-run relationship.

Distributed Lag Explanation

Another explanation of the short- and long-run relationships can be pro-vided by looking at the following distributed lag model:

(A6) In this case, the long-run (steady-state) relationship between y_tand x_t is represented by the coefficient g1 , which is obtained from (after

(1⫺ g2)

yt⫽ g0⫹ g1xt⫹ g2yt⫺1⫹ ut

g(w) ⫽ 冟1 ⫺ e⫺iw冟2_{⫽ 2(1 ⫺ cos(w))}

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dropping the time subscript):

(A7) On the other hand, the short-run relationship is represented by the coefﬁcient . We can derive the following equation from Equation (A6):

or

(A8) Comparison of Equation (A8) with Equation (A2), shows clearly that Equation (A2) represents a short-run relationship between the two series

xtand yt.

Co-Integration Explanation

If the two series x_t and y_t are nonstationary and co-integrated, then Equation (A1) is considered to be a long-run relationship in the sense that it is the stationary relationship between the nonstationary series. In this case, Equation (A1) is called the co-integrating regression. Therefore, from the viewpoint of co-integration analysis, Equation (A1) can be considered as a long-run relationship.

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